Author: UrsFormat: MarkdownItex> I have added a link to AGT correspondence,
Thanks.
> which I have extended in the meantime.
Which entry have you expanded? I am not sure if I see it.
> hope you will point out that some dualities in physics involve more than two
I already did, I thought. Maybe I should expand on it furhter.

I have added a link to AGT correspondence,

Thanks.

which I have extended in the meantime.

Which entry have you expanded? I am not sure if I see it.

hope you will point out that some dualities in physics involve more than two

Author: UrsFormat: MarkdownItex> >which I have extended in the meantime.
> Which entry have you expanded? I am not sure if I see it.
Ah, you haven't expanded _[[AGT correspondence]]_, you have expanded _[[AGT conjecture]]_!
I am merging the two entries now...

Author: zskodaFormat: MarkdownItexThat is strange. I thought there was only [[AGT conjecture]] page which I created on June 25 and reported. I have just added there the redirect to it in the moment when I referred to it in duality page. Now I see you created different entry on July 18 (search AGT). AGT conjecture is much more quoted phrase than AGT correspondence which I have seen just few times, but probably it will change in future, regarding that AGT conjecture is already proven in many important cases.

That is strange. I thought there was only AGT conjecture page which I created on June 25 and reported. I have just added there the redirect to it in the moment when I referred to it in duality page. Now I see you created different entry on July 18 (search AGT). AGT conjecture is much more quoted phrase than AGT correspondence which I have seen just few times, but probably it will change in future, regarding that AGT conjecture is already proven in many important cases.

Author: David_CorfieldFormat: MarkdownItexI see this [conference](http://dualitiesblog.wordpress.com/) I'm attending soon is mostly attended by people coming from the physics side. Since I'm coming from the mathematics side, it would be good to say a few words about what kind of mathematical duality (if any) is involved in certain cases.
So how's this?
* [[holographic duality]]: not really a duality. For a boundary theory, need there even be a dual bulk theory, and if so need it be unique?
* [[T-duality]]: exchange of data inducing equivalent QFTs. But we wouldn't call 2x3 the dual of 3x2 just because they both result in 6. Also, correspondence inducing integral transform isomorphism.
* [[Mirror symmetry]]: case of T-duality. Exchange of data induces equivalent QFTs but swaps A and B models. But then I wouldn't call reflection in $x = y$ such a big deal just because it swaps coordinates.
* [[S-duality]]: vestige of higher SL(2, Z) symmetry. But E-M duality related to Hodge duality from pairing.
Hmm, they're not like good old dual adjoint equivalences in maths, are they? Then again I guess one's happy enough to call projective duality a duality which is the consequence of an outer automorphism shuffling of a group which reorders the dimensions of the constitutent elements.

I see this conference I’m attending soon is mostly attended by people coming from the physics side. Since I’m coming from the mathematics side, it would be good to say a few words about what kind of mathematical duality (if any) is involved in certain cases.

So how’s this?

holographic duality: not really a duality. For a boundary theory, need there even be a dual bulk theory, and if so need it be unique?

T-duality: exchange of data inducing equivalent QFTs. But we wouldn’t call 2x3 the dual of 3x2 just because they both result in 6. Also, correspondence inducing integral transform isomorphism.

Mirror symmetry: case of T-duality. Exchange of data induces equivalent QFTs but swaps A and B models. But then I wouldn’t call reflection in x=yx = y such a big deal just because it swaps coordinates.

Hmm, they’re not like good old dual adjoint equivalences in maths, are they? Then again I guess one’s happy enough to call projective duality a duality which is the consequence of an outer automorphism shuffling of a group which reorders the dimensions of the constitutent elements.

Author: David_CorfieldFormat: MarkdownItexI see Urs has [already](http://golem.ph.utexas.edu/category/2011/05/making_things_simpler_by_duali.html#c037943) given an opinion:
>I think this is really all there is to it: in physics they discovered interesting equivalences and called them dualities.
E.g., mirror symmetry as an equivalence.
If that's the case, and since speakers are interested in what dual theories tell us about the way the world is, perhaps I could play a more useful role in expounding the HoTT approach to equivalence.

Author: UrsFormat: MarkdownItexYes, to a large extent the term "duality" in physics is used somewhat thoughtlessly.
There is [[electric-magnetic duality]] which, when it really does work, is a $\mathbb{Z}_2$-action on the parameter space of (super-)Yang-Mills theory and hence is close to justifiably be called a duality. Notice that this is thought to really be part of an $SL(2,\mathbb{Z})$-action, though, called [[S-duality]].
T-duality with its similarity to Fourier-Mukai duality (see at [[topological T-duality]]) comes pretty close to being a duality in the sense of mathematics. However, there are extra choices involved and indeed any given string background may have none, one _or more than one_ "T-duals".
As you notice, since mirror symmetry is by and large a version of T-duality, same comments would apply there, though here the situation is a bit more rigid with the action on the Hodge diamond really being a $\mathbb{Z}_2$-action.
The proliferation of dualities in physics came with the "second string revolution" in the 90s, when type II A type II B, type I, Het E, Het O, and Sugra 11 were all realized to be equivalent to each other, at least in certain limits. Since then one speaks of "duality in string theory", even though this web of equivalences is much more like a bunch of different coordinate charts on a big moduli space than anything that one would call "duality" in mathematics.
So, yes, I think if one grants different meaning of the same words across discipline borders, then then real point of "duality in physics" is that (local Lagrangian gauge) QFT is a pretty rich beast, and that its "moduli space" contains some obvious and some rather subtle equivalences. The more subtle such an equivalence is, the more likely is it going to be called a "duality".

Yes, to a large extent the term “duality” in physics is used somewhat thoughtlessly.

There is electric-magnetic duality which, when it really does work, is a &Zopf;2\mathbb{Z}_2-action on the parameter space of (super-)Yang-Mills theory and hence is close to justifiably be called a duality. Notice that this is thought to really be part of an SL(2,&Zopf;)SL(2,\mathbb{Z})-action, though, called S-duality.

T-duality with its similarity to Fourier-Mukai duality (see at topological T-duality) comes pretty close to being a duality in the sense of mathematics. However, there are extra choices involved and indeed any given string background may have none, one or more than one “T-duals”.

As you notice, since mirror symmetry is by and large a version of T-duality, same comments would apply there, though here the situation is a bit more rigid with the action on the Hodge diamond really being a &Zopf;2\mathbb{Z}_2-action.

The proliferation of dualities in physics came with the “second string revolution” in the 90s, when type II A type II B, type I, Het E, Het O, and Sugra 11 were all realized to be equivalent to each other, at least in certain limits. Since then one speaks of “duality in string theory”, even though this web of equivalences is much more like a bunch of different coordinate charts on a big moduli space than anything that one would call “duality” in mathematics.

So, yes, I think if one grants different meaning of the same words across discipline borders, then then real point of “duality in physics” is that (local Lagrangian gauge) QFT is a pretty rich beast, and that its “moduli space” contains some obvious and some rather subtle equivalences. The more subtle such an equivalence is, the more likely is it going to be called a “duality”.

Author: DavidRobertsFormat: MarkdownItexVarghese Mathai and collaborators have released a [couple](http://arxiv.org/abs/1405.5844) of [papers](http://arxiv.org/abs/1409.1296) on 'spherical T-duality', really a variant of the usual with the following replacements: $S^1 = U(1) \mapsto S^3=SU(2)$, 3-form on total space of bundle $\mapsto$ 7-form (or rather, cocycles). The subtitle of the second paper is telling: _An infinity of spherical T-duals for non-principal SU(2)-bundles_. Really there's some sort of groupoid of duals, with some as-yet unidentified automorphism group.

Varghese Mathai and collaborators have released a couple of papers on ’spherical T-duality’, really a variant of the usual with the following replacements: S1=U(1)&map;S3=SU(2)S^1 = U(1) \mapsto S^3=SU(2), 3-form on total space of bundle &map;\mapsto 7-form (or rather, cocycles). The subtitle of the second paper is telling: An infinity of spherical T-duals for non-principal SU(2)-bundles. Really there’s some sort of groupoid of duals, with some as-yet unidentified automorphism group.

Author: UrsFormat: MarkdownItexMaybe one should keep in mind other instances of misleadingly similar but actually different use of terminology across the math/physics border. For instance when physicists speak of "covariance" they mean something that may with some effort be traced back to what would be called "covariance" in mathematics, but which really has a meaning of its own.
Other instances of such an almost similarities is "model", "theory" and probably many more. As always, one should pay attention to the difference between what somebody says, and what he or she actually means.

Maybe one should keep in mind other instances of misleadingly similar but actually different use of terminology across the math/physics border. For instance when physicists speak of “covariance” they mean something that may with some effort be traced back to what would be called “covariance” in mathematics, but which really has a meaning of its own.

Other instances of such an almost similarities is “model”, “theory” and probably many more. As always, one should pay attention to the difference between what somebody says, and what he or she actually means.

Author: UrsFormat: MarkdownItexAll that said, it should maybe be amplified that of course the genuine duality in the sense of mathematics (dualizing objects) also plays a huge role in physics, even if it is not what the physicist thinks about first when saying "duality".
For instance Dirac's famous bra-ket notation formalism which has become the hallmark of much of quantum physics is of course a formalism for dealing with dualizing objects. And via the theory of cobordism categories we know that key concepts such as traces producing partition functions etc. are based on the existence of dual objects. Dualizing objects play a pivotal role in quantization, often implicitly so, but very explicitly for instance in what I wrote at "Quantization via Linear homotopy types".

All that said, it should maybe be amplified that of course the genuine duality in the sense of mathematics (dualizing objects) also plays a huge role in physics, even if it is not what the physicist thinks about first when saying “duality”.

For instance Dirac’s famous bra-ket notation formalism which has become the hallmark of much of quantum physics is of course a formalism for dealing with dualizing objects. And via the theory of cobordism categories we know that key concepts such as traces producing partition functions etc. are based on the existence of dual objects. Dualizing objects play a pivotal role in quantization, often implicitly so, but very explicitly for instance in what I wrote at “Quantization via Linear homotopy types”.

Author: UrsFormat: MarkdownItexMeanwhile David and I have been chatting in person more on the issue of what it means mathematically when one speaks of [[duality in physics]]. What I had suggested to David in person I have now briefly recorded in the entry in a new section _[Formalization](http://ncatlab.org/nlab/show/duality+in+physics#Formalization)_.

Meanwhile David and I have been chatting in person more on the issue of what it means mathematically when one speaks of duality in physics. What I had suggested to David in person I have now briefly recorded in the entry in a new section Formalization.

Author: David_CorfieldFormat: MarkdownItexGreat you're adding this, and fun to have chatted about it in Paris. It seems I have a further month to add things to the final draft of the paper I'm writing on duality, so it would be very good to continue. There's something I haven't quite grasped yet about groupoid objects and effective epimorphisms, etc. But post-Paris I have some catching up to do, so haven't time just now to see what it is I'm missing.
We should also remember some of the other questions we discussed:
* Could one notice $\hbar$ in classical mechanics? Would this need a compact phase space?
* How would one describe the rise of quantum mechanics had one had the presentation of classical physics done &agrave; la Urs.
* Riemannian geometry has 2 descriptions, as Cartan geometry for Euclidean space, and in terms of reduction of structure group $GL(n)$ to $O(n)$. What’s special about the Euclidean-Riemannian relation that allows this, where perhaps Cartan geometries for other Klein geometries don’t?
* Related to this thread, could one give a HoTT account of T-duality for QFT rather like that for covariance in GRT?
There may have been others.

Great you’re adding this, and fun to have chatted about it in Paris. It seems I have a further month to add things to the final draft of the paper I’m writing on duality, so it would be very good to continue. There’s something I haven’t quite grasped yet about groupoid objects and effective epimorphisms, etc. But post-Paris I have some catching up to do, so haven’t time just now to see what it is I’m missing.

We should also remember some of the other questions we discussed:

Could one notice &hbar;\hbar in classical mechanics? Would this need a compact phase space?

How would one describe the rise of quantum mechanics had one had the presentation of classical physics done à la Urs.

Riemannian geometry has 2 descriptions, as Cartan geometry for Euclidean space, and in terms of reduction of structure group GL(n)GL(n) to O(n)O(n). What’s special about the Euclidean-Riemannian relation that allows this, where perhaps Cartan geometries for other Klein geometries don’t?

Related to this thread, could one give a HoTT account of T-duality for QFT rather like that for covariance in GRT?

Author: UrsFormat: MarkdownItexre #16: thanks for compiling that list. Yes, I will try to get back to all of these items once I am not under insane time pressure anymore. (That'll be by next week).
re #17: yes, while talking to David Corfield it occurrred to be that this may just the be good way to say it. It's excellent how talking with David C. brings out all these points. You would think that it would have bee the task of mathematical physicists to ask what the formal definition of such dualities is. But they didn't, only David C. did. We were talking about this over lunch and it so happened that next to us is an expert on mathematical mirror symmetry. But he hadn't thought about it either.

re #16: thanks for compiling that list. Yes, I will try to get back to all of these items once I am not under insane time pressure anymore. (That’ll be by next week).

re #17: yes, while talking to David Corfield it occurrred to be that this may just the be good way to say it. It’s excellent how talking with David C. brings out all these points. You would think that it would have bee the task of mathematical physicists to ask what the formal definition of such dualities is. But they didn’t, only David C. did. We were talking about this over lunch and it so happened that next to us is an expert on mathematical mirror symmetry. But he hadn’t thought about it either.

Author: UrsFormat: MarkdownItexone more,but then I really need to run, regarding item #1: while not a full answer, at least one thing to mention are the [[classical anomalies]] an example of which (see there) appears already in plain Galailean mechanics.

one more,but then I really need to run, regarding item #1: while not a full answer, at least one thing to mention are the classical anomalies an example of which (see there) appears already in plain Galailean mechanics.

Author: David_CorfieldFormat: MarkdownItexRe #20, you mean mass? We had a discussion about that back at the [cafe](https://golem.ph.utexas.edu/category/2012/06/cohomology_in_everyday_life.html#c041693):
>DC: I see Santiago García in Hidden invariance of the free classical particle writes that mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” Is this an interesting point of view?
>JB: Yes! The locus classicus for this viewpoint may be Guillemin and Sternberg’s book Symplectic Techniques in Physics.
>Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.
>Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.
>Working out how the last two paragraphs are related is a fun exercise in taking some ideas from cohomology and seeing what they amount to in ‘real life’.
Hmm, what John's saying about the difference between classical and mechanics, the Planck constant, is the very issue that started off my query in that it seemed to have already appeared according to the account of classical mechanics you gave last week in Paris.
Then you add
>US: In slight disguise, one can see this cocycle also controls already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the “WZW term” that however comes down to be the ordinary free action.

Re #20, you mean mass? We had a discussion about that back at the cafe:

DC: I see Santiago García in Hidden invariance of the free classical particle writes that mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” Is this an interesting point of view?

JB: Yes! The locus classicus for this viewpoint may be Guillemin and Sternberg’s book Symplectic Techniques in Physics.

Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.

Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.

Working out how the last two paragraphs are related is a fun exercise in taking some ideas from cohomology and seeing what they amount to in ‘real life’.

Hmm, what John’s saying about the difference between classical and mechanics, the Planck constant, is the very issue that started off my query in that it seemed to have already appeared according to the account of classical mechanics you gave last week in Paris.

Then you add

US: In slight disguise, one can see this cocycle also controls already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the “WZW term” that however comes down to be the ordinary free action.

Author: David_CorfieldFormat: MarkdownItexRe #15, when you have time to come back to this, at [Formalization](http://ncatlab.org/nlab/show/duality+in+physics#Formalization), the next step up from dualities in the diagram with three arrows down to it is what? A triple of Lagrangians and equivalences between their QFTs? What does such a tower give us?

Re #15, when you have time to come back to this, at Formalization, the next step up from dualities in the diagram with three arrows down to it is what? A triple of Lagrangians and equivalences between their QFTs? What does such a tower give us?

Author: UrsFormat: MarkdownItexYes, this continues in the way that Cech covers do (whence the name): above "$Dualities$" there is the space of triples of Lagrangian data that all have the same quantization, equipped with dualities between any two of them, and equipped with an equivalence of dualities between the composite of two of these and the third:
$$
\array{
&& \vdots
\\
&& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \underset{LagrangianQFT}{\times} LagrangianDat
\\
&& \downarrow \downarrow \downarrow
\\
Dualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData
\\
&& \downarrow \downarrow
\\
&& LagrangianData
}
$$
Such towers are to be thought of as the incarnation of equivalence relations in homotopy logic. A plain equivalence relation is just the first stage of such a tower
$$
\array{
Dualities
\\
\downarrow \downarrow
\\
Lagrangians
}
$$
The conditions on an equivalence relation (reflexivity, transitivity, symmetry) may be read as those on a groupoid object (identity, composition, inverses). So now in homotopy logic this is boosted to an $\infty$-groupoid object by relaxing all three to hold only up to higher coherent homotopies.
The bottom-most arrow
$$
\array{
Lagrangians
\\
\downarrow
\\
LagrangianQFTs
}
$$
is the quotient projection of the equivalence relation. In 1-logic this would be its "cokernel", here in homotopy logic it is the homotopy colimit over the full simplicial diagram.
So the perspective of the full diagram somehow gives the usual way of speaking in QFT also a reverse:
instead of saying
a) that two Lagrangians are dual if there is an equivalence between the QFTs which they induce under quantization,
we may turn this around and say that therefore
b) quantization is the result of forming the homotopy quotient of the space of Lagrangian data by these duality relations.
It is one of the clauses of the [[Giraud theorem]] in $\infty$-topos theory that these two perspectives are equivalent.

Yes, this continues in the way that Cech covers do (whence the name): above “DualitiesDualities” there is the space of triples of Lagrangian data that all have the same quantization, equipped with dualities between any two of them, and equipped with an equivalence of dualities between the composite of two of these and the third:

The conditions on an equivalence relation (reflexivity, transitivity, symmetry) may be read as those on a groupoid object (identity, composition, inverses). So now in homotopy logic this is boosted to an ∞\infty-groupoid object by relaxing all three to hold only up to higher coherent homotopies.

Author: UrsFormat: MarkdownItexI have edited a tad more at _[Formalization and Relation to mathematical duality](http://ncatlab.org/nlab/show/duality+in+physics#Formalization)_ in order to streamline more.

Author: David_CorfieldFormat: MarkdownItexThe only mention of three things being related in this area that I can recall is Frenkel in [Gauge Theory and Langlands Duality](http://arxiv.org/abs/0906.2747) on p. 18 relating A-Branes on something to B-branes on something else to D-modules on yet something else. He says there that the dualities along each side are Homological Mirror Symmetry, categorical Langlands correspondence, and an unnamed one worked on by Witten and Kapustin, and Nadler and Zaslow.
Beyond simple $Z_2$ actions there is also the SL(2, Z) symmetry of S-duality. Then there's T-duality where something may have #10
>none, one or more than one “T-duals”,
as also appears to be the case with the spherical T-duals appearing in the papers David R. mentions above in #11.
I wonder how these cases could be formalised.

The only mention of three things being related in this area that I can recall is Frenkel in Gauge Theory and Langlands Duality on p. 18 relating A-Branes on something to B-branes on something else to D-modules on yet something else. He says there that the dualities along each side are Homological Mirror Symmetry, categorical Langlands correspondence, and an unnamed one worked on by Witten and Kapustin, and Nadler and Zaslow.

Beyond simple Z2Z_2 actions there is also the SL(2, Z) symmetry of S-duality. Then there’s T-duality where something may have #10

none, one or more than one “T-duals”,

as also appears to be the case with the spherical T-duals appearing in the papers David R. mentions above in #11.

Author: UrsFormat: MarkdownItexThat there are triples appearing here is mostly an artifact of the simplicial language. What counts really is that there are 2-morphisms involved, namely 2-equivalences of the form
$$
\array{
&& L_b
\\
& \nearrow &\Downarrow& \searrow
\\
L_a && \longrightarrow && L_c
}
$$
We may consider just the special case that $L_b \to L_c$ is the identity, then this reduces to the more vivid globular 2-cell
$$
\array{
& \nearrow \searrow
\\
L_a & \Downarrow & L_c
\\
& \searrow \nearrow
}
$$
So these are "dualities of dualities" in that they are equivalences between equivalences between quantizations of Lagrangians.
Similarly the $n$th stage in the Cech nerve of the map $quantization : LagrangianData \to LagrangianQFT$ is only superficially about (n+1)-tuples of Lagrangian all dual to each other, while its real content is that it exhibits $n$-fold higher order dualities.
Notice that we already captured the aspect that "duality" in physics is not something that goes back and forth between just two objects when realizing that it's really an $\infty$-groupoid object. This takes care of any numbers of objects being equivalent to each other. But we discovered something new, too, that is not in the literature, namely that there are also higher order dualities.

That there are triples appearing here is mostly an artifact of the simplicial language. What counts really is that there are 2-morphisms involved, namely 2-equivalences of the form

So these are “dualities of dualities” in that they are equivalences between equivalences between quantizations of Lagrangians.

Similarly the nnth stage in the Cech nerve of the map quantization:LagrangianData&rightarrow;LagrangianQFTquantization : LagrangianData \to LagrangianQFT is only superficially about (n+1)-tuples of Lagrangian all dual to each other, while its real content is that it exhibits nn-fold higher order dualities.

Notice that we already captured the aspect that “duality” in physics is not something that goes back and forth between just two objects when realizing that it’s really an ∞\infty-groupoid object. This takes care of any numbers of objects being equivalent to each other. But we discovered something new, too, that is not in the literature, namely that there are also higher order dualities.

Author: UrsFormat: MarkdownItexYes, absoluteley, one would be inclined to ask if that triangle on p. 18 of [arXiv:0906.2747](http://arxiv.org/abs/0906.2747) may be filled by a 2-morphism! If so, this would be a "duality of dualities" in the sense we are discussing here.
What I meant to highlight is that this question appears more generally, independently of whether there is a triangle of equivalences -- though of course that is an excellent example to look at.

Yes, absoluteley, one would be inclined to ask if that triangle on p. 18 of arXiv:0906.2747 may be filled by a 2-morphism! If so, this would be a “duality of dualities” in the sense we are discussing here.

What I meant to highlight is that this question appears more generally, independently of whether there is a triangle of equivalences – though of course that is an excellent example to look at.

Author: UrsFormat: MarkdownItexThe construction that we are looking at will consider all 2-equivalences that are present. We have heard over lunch the other day that not much is known about which 1-morphisms actually implement the mirror symmetry between two Calabi-Yau $A_\infty$-categories. Suppose somebody constructs any one, then the immediate candidates for interesting 2-morphisms to consider are the auto-equivalences of that 1-morphism. Next, sombody else constructs another one, then either it is equivalent to the previous one or not. If it is, there is another 2-equivalence, and so forth.
Does Frenkel or anyone make any comment on whether that triangle is suppose to commute, up to equivalence?

The construction that we are looking at will consider all 2-equivalences that are present. We have heard over lunch the other day that not much is known about which 1-morphisms actually implement the mirror symmetry between two Calabi-Yau A∞A_\infty-categories. Suppose somebody constructs any one, then the immediate candidates for interesting 2-morphisms to consider are the auto-equivalences of that 1-morphism. Next, sombody else constructs another one, then either it is equivalent to the previous one or not. If it is, there is another 2-equivalence, and so forth.

Does Frenkel or anyone make any comment on whether that triangle is suppose to commute, up to equivalence?

Author: David_CorfieldFormat: MarkdownItexLet's see.
>Thus, according to Kapustin and Witten, the (categorical) geometric Langlands correspondence (3.1) may be obtained in two steps. The first step is the Homological Mirror Symmetry (5.2) of the Hitchin moduli spaces for two dual groups, and the second step is the above link between the A-branes and D-modules.
So that's getting at the B-brane/D-module correspondence via the other two sides, i.e., using A-branes to mediate.
But also,
>There is actually more structure in the triangle (5.3). On each of these three categories we have an action of certain functors, and all equivalences between them are supposed to commute with these functors.

Let’s see.

Thus, according to Kapustin and Witten, the (categorical) geometric Langlands correspondence (3.1) may be obtained in two steps. The first step is the Homological Mirror Symmetry (5.2) of the Hitchin moduli spaces for two dual groups, and the second step is the above link between the A-branes and D-modules.

So that’s getting at the B-brane/D-module correspondence via the other two sides, i.e., using A-branes to mediate.

But also,

There is actually more structure in the triangle (5.3). On each of these three categories we have an action of certain functors, and all equivalences between them are supposed to commute with these functors.

Author: UrsFormat: MarkdownItexThe second comment is, I think, about everything respecting the action of Hecke operators or similar, on top of being a suitable equivalence.
I would tend to go hunting for explicit constructions of (just) homological mirror symmetry as equivalences of $A_\infty$-categories. I'll see if I find something...

The second comment is, I think, about everything respecting the action of Hecke operators or similar, on top of being a suitable equivalence.

I would tend to go hunting for explicit constructions of (just) homological mirror symmetry as equivalences of A∞A_\infty-categories. I’ll see if I find something…

Author: David_CorfieldFormat: MarkdownItexIn [Formalization](http://ncatlab.org/nlab/show/duality+in+physics#Formalization)
>So now in homotopy logic this is boosted to an groupoid object in an (∞,1)-category
Did you mean 'an $\infty$-groupoid object'?

Author: David_CorfieldFormat: MarkdownItexOK. I was going to change it, but I see it points to [[groupoid object in an (infinity,1)-category]]. There it says
>By the logic of vertical categorification, an internal ∞-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category C with (∞,1)-pullbacks.
So one does drop the $\infty$ when speaking of what kind of object is internal?

Author: UrsFormat: MarkdownItexYes, the default meaning of "groupoid" internal to an $\infty$-topos is that which externally is an $\infty$-groupoid. If one actually means a 1-groupoid one should speak of a truncated groupoid or the like.
This is in the spirit of homotopy logic: if we say "groupoid" in the internal language then it comes out as an $\infty$-groupoid. Well, at least up to the point where one says "associativity" internally, beyond that the coherence for associativity is something one might forget to say. It is presently an open problem how to bring higher coherences systematically into an internal homotopy logic such as HoTT.

Yes, the default meaning of “groupoid” internal to an ∞\infty-topos is that which externally is an ∞\infty-groupoid. If one actually means a 1-groupoid one should speak of a truncated groupoid or the like.

This is in the spirit of homotopy logic: if we say “groupoid” in the internal language then it comes out as an ∞\infty-groupoid. Well, at least up to the point where one says “associativity” internally, beyond that the coherence for associativity is something one might forget to say. It is presently an open problem how to bring higher coherences systematically into an internal homotopy logic such as HoTT.

Author: David_CorfieldFormat: MarkdownItexAlong with finding examples of things in the higher levels of the diagram, such as dualities of dualities, there's the philosopher's question of which level is the 'real' one. Of course, they're just working with the Lagrangian Data and QFT levels. The perplexity, if you remember, comes from one QFT arising from two or more Lagrangians which appear to say very different things about the world.
What should we say now? Presumably that Lagrangian Data level is not privileged in any way. Is it that in HoTT spirit, we'll end up saying something very simple, but which has a complicated external interpretation, such as when 'There is a path' becomes 'There is an object of paths with superposed actions, up to higher-gauge equivalence,...'
Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?

Along with finding examples of things in the higher levels of the diagram, such as dualities of dualities, there’s the philosopher’s question of which level is the ’real’ one. Of course, they’re just working with the Lagrangian Data and QFT levels. The perplexity, if you remember, comes from one QFT arising from two or more Lagrangians which appear to say very different things about the world.

What should we say now? Presumably that Lagrangian Data level is not privileged in any way. Is it that in HoTT spirit, we’ll end up saying something very simple, but which has a complicated external interpretation, such as when ’There is a path’ becomes ’There is an object of paths with superposed actions, up to higher-gauge equivalence,…’

Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?

Author: UrsFormat: MarkdownItexIt seems to me that this is much unexplored, and I am not aware of anyone besides you and me who has really raised this issue.
My whole point with "local pre-quantum field theory" etc. is that the Lagrangian data may well matter, in that just quantizED field theories are not an accurate description of reality, but that the information of what they are the quantization of is part of what it needs to have a model in physics. There are lots of indications that this is so, but when it comes to "saying things about the world" of course we are in deep water. It's my long term strategy to develop some formal foundations of physics to the point where we may talk about this with more confidence, but maybe that still needs more work.
I'd just generally be careful with making naive identifications between the symols that we bring to paper and "the world". When considering dualites of the kind we are considering here, then we are talking about concepts of fundamental physics at length scales many orders of magnitude smaller than what present day accelerators may see. It seems wrong to worry much about whether on that scale there "really is" that Calabi-Yau manifold of internal degrees of freedom or rather its mirror dual, because there is no reason to expect that the concept "spacetime manifold" by itself is of much meaning at that scale. I would turn this around and think that the duality between descriptions of reality by such manifolds is exactly what characterizes the actual reality at such tiny scales. Much like the "wave-particle duality" of plain quantum physics is not something where we are to choose between one picture or the other to get a picture of the world, but rather is a reflection of the fact that only some fusion of both pictures is the real picture.
The concept of [[T-fold]] gets to this point a bit more. Worrying for a T-fold whether it locally "really" is that torus bundle or rather its T-dual seems to be the same kind of wrong worry that our ancestors had when looking at points in a manifold acted on by diffeomorphisms and finding a "hole paradox".

It seems to me that this is much unexplored, and I am not aware of anyone besides you and me who has really raised this issue.

My whole point with “local pre-quantum field theory” etc. is that the Lagrangian data may well matter, in that just quantizED field theories are not an accurate description of reality, but that the information of what they are the quantization of is part of what it needs to have a model in physics. There are lots of indications that this is so, but when it comes to “saying things about the world” of course we are in deep water. It’s my long term strategy to develop some formal foundations of physics to the point where we may talk about this with more confidence, but maybe that still needs more work.

I’d just generally be careful with making naive identifications between the symols that we bring to paper and “the world”. When considering dualites of the kind we are considering here, then we are talking about concepts of fundamental physics at length scales many orders of magnitude smaller than what present day accelerators may see. It seems wrong to worry much about whether on that scale there “really is” that Calabi-Yau manifold of internal degrees of freedom or rather its mirror dual, because there is no reason to expect that the concept “spacetime manifold” by itself is of much meaning at that scale. I would turn this around and think that the duality between descriptions of reality by such manifolds is exactly what characterizes the actual reality at such tiny scales. Much like the “wave-particle duality” of plain quantum physics is not something where we are to choose between one picture or the other to get a picture of the world, but rather is a reflection of the fact that only some fusion of both pictures is the real picture.

The concept of T-fold gets to this point a bit more. Worrying for a T-fold whether it locally “really” is that torus bundle or rather its T-dual seems to be the same kind of wrong worry that our ancestors had when looking at points in a manifold acted on by diffeomorphisms and finding a “hole paradox”.

Author: UrsFormat: MarkdownItex> Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?
The trouble is that T-duality is about "geometric" strings described by genuine 2d CFT, and for these much less is known than for their topological counterparts. For the case of homological mirror symmetry between topological string models we have a very clear mathematical picture of what that map $quantization \colon LagrangianData \to LagrangianQFTs$ is, as a map of moduli spaces (namely the one from CY manifolds to their CY $A_\infty$-categories), but for genuine CFTs nothing of this mathematical precision is in reach. Most of what is understood about T-duality mathematically is accordingly [[topological T-duality]] which is really a bit of educated guesswork for extracting the underlying math from fully fledged T-duality.

Back to T-duality, is there any way we can see T-folds and T-dual 2-group relating to the tower?

The trouble is that T-duality is about “geometric” strings described by genuine 2d CFT, and for these much less is known than for their topological counterparts. For the case of homological mirror symmetry between topological string models we have a very clear mathematical picture of what that map quantization&colon;LagrangianData&rightarrow;LagrangianQFTsquantization \colon LagrangianData \to LagrangianQFTs is, as a map of moduli spaces (namely the one from CY manifolds to their CY A∞A_\infty-categories), but for genuine CFTs nothing of this mathematical precision is in reach. Most of what is understood about T-duality mathematically is accordingly topological T-duality which is really a bit of educated guesswork for extracting the underlying math from fully fledged T-duality.

Author: David_CorfieldFormat: MarkdownItex>...a map of moduli spaces (namely the one from CY manifolds to their CY $A_\infty$-categories)
I thought it was the case that there were two maps from CY manifolds to $A_{\infty}$-categories, such as [here](http://users2.unimi.it/stellari/Research/Papers/SurvGenK3.pdf):
> for two Calabi–Yau manifolds X and Y which are mirror, the bounded derived category of coherent sheaves on one variety should be equivalent to the derived Fukaya category of the other one.
Oh but at [[mirror symmetry]] you factor these maps via $X \mapsto SCFT(X)$, then different maps go out from $SCFT(X)$ to its A and B models. Hmm, how does that fit in with the tower diagram?

…a map of moduli spaces (namely the one from CY manifolds to their CY A∞A_\infty-categories)

I thought it was the case that there were two maps from CY manifolds to A∞A_{\infty}-categories, such as here:

for two Calabi–Yau manifolds X and Y which are mirror, the bounded derived category of coherent sheaves on one variety should be equivalent to the derived Fukaya category of the other one.

Oh but at mirror symmetry you factor these maps via X&map;SCFT(X)X \mapsto SCFT(X), then different maps go out from SCFT(X)SCFT(X) to its A and B models. Hmm, how does that fit in with the tower diagram?

Author: UrsFormat: MarkdownItexRight, the "Lagrangian data" here is that of a certain type of sigma-model with target space a given Calabi-Yau manifold. The types of sigma models here determine the actual Lagrangian built from the CY. For the present purpose one should say that a point in "$LagrangianData$" is a pair consisting of a CY manifold and an element in $\{A,B\}$ indicating which of two possible Lagrangians we build from this.

Right, the “Lagrangian data” here is that of a certain type of sigma-model with target space a given Calabi-Yau manifold. The types of sigma models here determine the actual Lagrangian built from the CY. For the present purpose one should say that a point in “LagrangianDataLagrangianData” is a pair consisting of a CY manifold and an element in {A,B}\{A,B\} indicating which of two possible Lagrangians we build from this.

Author: David_CorfieldFormat: MarkdownItexWhere in #23 you say
>Such towers are to be thought of as the incarnation of equivalence relations in homotopy logic,
is there a formulation of this construction in the internal logic?

Where in #23 you say

Such towers are to be thought of as the incarnation of equivalence relations in homotopy logic,

Author: UrsFormat: MarkdownItexYes... except possibly for the fact mentioned elsehwere recently, that higher coherences are presently hard so say all the way to infinity in the internal logic.
But in low stages the idea of a groupoid object is just that embodied in univalence: there on the type $Type$ one considers the equivalence relation $\underset{X,Y \colon Type}{\sum} Equiv_{Type}(X,Y)$ and the univalence axiom in this case comes down to saying that this is the equivalence already encoded by the morphisms ("paths") inside $Type$, hence that $Type$ is already itself the (homotopy) quotient under this relation.
Only that actually the full concept of _homotopy quotient_ is presently not expressed in HoTT, because taken at face value it is the colimit over a simplicial diagram, and it is problematic to encode these.
On the other hand, if we are happy with using the statement of the Giraud theorem that holds externally in $\infty$-toposes, then we know that the simplicial description of groupoid objects is equivalently encoded in the [[1-epimorphism]] which is their homotopy quotient map, and "1-epimorphism" one may easily say in HoTT. From this perspective saying (internally) that
$$
quantization \colon LagrangianData \longrightarrow LagrangianQFT
$$
is a 1-epimorphism already is equivalent to saying that there is an equivalence relation up to coherent homotopy on $LagrangianData$ given by dualities.
This may seem like a cheap trick, but I actually think this is a useful perspective. Something similar happens with the classification of principal $\infty$-bundles and with [[internal (co-)limits]]: externally the theory in both cases a priori needs homotopy colimits over non-finite diagrams, but since these turn out to be equivalent to something internal and "elementary", we may just take that elementary characterization as the defining one and happily proceed internally.
And indeed, sticking to the internal logic tends to make this more true than it might seem from the external perspective. For instance here the fact that any 1-epimorphism $X \to Q$ may be regarded as an equivalence relation becomes manifest internally once one regards it as exhibiting $X$ as a $Q$-dependent type. Then one may ask what $X(q)$ is in the context of a term $q \colon Q$ and one finds that $X(q)$ is the homotopy fiber of the morphism over $q$ hence is the elements of $X$ together with the $\infty$-equivalence relations between them as encoded by the original 1-epimorphism.
And hence we find as a theorem in the internal logic:
> Duality is the equivalence of Lagrangians in the context of a joint quantization.
:-)

Yes… except possibly for the fact mentioned elsehwere recently, that higher coherences are presently hard so say all the way to infinity in the internal logic.

But in low stages the idea of a groupoid object is just that embodied in univalence: there on the type TypeType one considers the equivalence relation &Sum;X,Y&colon;TypeEquivType(X,Y)\underset{X,Y \colon Type}{\sum} Equiv_{Type}(X,Y) and the univalence axiom in this case comes down to saying that this is the equivalence already encoded by the morphisms (“paths”) inside TypeType, hence that TypeType is already itself the (homotopy) quotient under this relation.

Only that actually the full concept of homotopy quotient is presently not expressed in HoTT, because taken at face value it is the colimit over a simplicial diagram, and it is problematic to encode these.

On the other hand, if we are happy with using the statement of the Giraud theorem that holds externally in ∞\infty-toposes, then we know that the simplicial description of groupoid objects is equivalently encoded in the 1-epimorphism which is their homotopy quotient map, and “1-epimorphism” one may easily say in HoTT. From this perspective saying (internally) that

is a 1-epimorphism already is equivalent to saying that there is an equivalence relation up to coherent homotopy on LagrangianDataLagrangianData given by dualities.

This may seem like a cheap trick, but I actually think this is a useful perspective. Something similar happens with the classification of principal ∞\infty-bundles and with internal (co-)limits: externally the theory in both cases a priori needs homotopy colimits over non-finite diagrams, but since these turn out to be equivalent to something internal and “elementary”, we may just take that elementary characterization as the defining one and happily proceed internally.

And indeed, sticking to the internal logic tends to make this more true than it might seem from the external perspective. For instance here the fact that any 1-epimorphism X&rightarrow;QX \to Q may be regarded as an equivalence relation becomes manifest internally once one regards it as exhibiting XX as a QQ-dependent type. Then one may ask what X(q)X(q) is in the context of a term q&colon;Qq \colon Q and one finds that X(q)X(q) is the homotopy fiber of the morphism over qq hence is the elements of XX together with the ∞\infty-equivalence relations between them as encoded by the original 1-epimorphism.

And hence we find as a theorem in the internal logic:

Duality is the equivalence of Lagrangians in the context of a joint quantization.

Author: David_CorfieldFormat: MarkdownItex>This may seem like a cheap trick, but I actually think this is a useful perspective.
That's the nub of it. What's the perspective that will continue to hold that this is a cheap trick, whatever you go on to say, because of some principles which, say, would require the expression of that colimit?
And conversely, can we understand your perspective to be more than just 'useful', but getting things 'right'?

This may seem like a cheap trick, but I actually think this is a useful perspective.

That’s the nub of it. What’s the perspective that will continue to hold that this is a cheap trick, whatever you go on to say, because of some principles which, say, would require the expression of that colimit?

And conversely, can we understand your perspective to be more than just ’useful’, but getting things ’right’?

Author: UrsFormat: MarkdownItexThe perspective that an equivalence relation is a 1-epimorphism is close to the traditional naive idea: the map says which elements sit in the same equivalence class. It seems to me that this is plausible to every kid, while the definition as an $\infty$-groupoid object in comparison is rather wild. What we learn in the lack of internal simplicial diagrams is that many things we do with them are equivalent to elementary operations that are much easier.
By the way, it is generally thought that it ought to be possible to express simplicial types and simplicial colimits in HoTT, just that due to some technicalities it hasn't been done yet. It's one of the most annoying gaps in present HoTT. But maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simlicial constructions.

The perspective that an equivalence relation is a 1-epimorphism is close to the traditional naive idea: the map says which elements sit in the same equivalence class. It seems to me that this is plausible to every kid, while the definition as an ∞\infty-groupoid object in comparison is rather wild. What we learn in the lack of internal simplicial diagrams is that many things we do with them are equivalent to elementary operations that are much easier.

By the way, it is generally thought that it ought to be possible to express simplicial types and simplicial colimits in HoTT, just that due to some technicalities it hasn’t been done yet. It’s one of the most annoying gaps in present HoTT. But maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simlicial constructions.

Author: UrsFormat: MarkdownItexSo on reflection, that would be my suggestion for how to present the situation in the most evident way in internal logic:
start with the 1-epi $quantization \colon LagrangianData \to LagrangianQFT$;
then "dicover" $n$-fold dualities for any finite $n$ by forming the $(n+1)$-fold homotopy fiber product of that with itself. Which is just the $(n+1)$-fold product of $LagrangianData$ with itself, regarded in the context of $LagrangianQFTs$.
One won't get the full simplicial diagram of higher dualities this way, but in fact that's not what most physicists or philosophers would expect to see or want to see anyway. It's maybe not really close to "elementary" thinking. Nevertheless, one discovers all the higher dualities to any degree and one is guaranteed, by the original map being a 1-epi, that everything is "right" from a mathematical point of view, too.

So on reflection, that would be my suggestion for how to present the situation in the most evident way in internal logic:

then “dicover” nn-fold dualities for any finite nn by forming the (n+1)(n+1)-fold homotopy fiber product of that with itself. Which is just the (n+1)(n+1)-fold product of LagrangianDataLagrangianData with itself, regarded in the context of LagrangianQFTsLagrangianQFTs.

One won’t get the full simplicial diagram of higher dualities this way, but in fact that’s not what most physicists or philosophers would expect to see or want to see anyway. It’s maybe not really close to “elementary” thinking. Nevertheless, one discovers all the higher dualities to any degree and one is guaranteed, by the original map being a 1-epi, that everything is “right” from a mathematical point of view, too.

Author: Mike ShulmanFormat: MarkdownItex> maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simplicial constructions.
An interesting point, that perhaps one of the things HoTT (and our current inability to deal with &infin;-coherences therein) teaches us is to avoid higher homotopy coherences whenever possible.
Of course, now I can hear my advisor saying "we knew that decades ago!"...

maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simplicial constructions.

An interesting point, that perhaps one of the things HoTT (and our current inability to deal with ∞-coherences therein) teaches us is to avoid higher homotopy coherences whenever possible.

Of course, now I can hear my advisor saying “we knew that decades ago!”…

Author: UrsFormat: MarkdownItexadded pointer to the new preprint [Polchinski14](http://ncatlab.org/nlab/show/duality+in+physics#Polchinski14) (which I suppose is to go along with David's contribution into the proceedings of the meeting that David is writing his contibution for, too)

added pointer to the new preprint Polchinski14 (which I suppose is to go along with David’s contribution into the proceedings of the meeting that David is writing his contibution for, too)

Author: David_CorfieldFormat: MarkdownItexYes, that's right. I included a brief mention of those ideas on effective epimorphisms from Lagrangian data. We'll have to spot a duality between dualities one day.

Yes, that’s right. I included a brief mention of those ideas on effective epimorphisms from Lagrangian data. We’ll have to spot a duality between dualities one day.

Author: UrsFormat: MarkdownItex> We’ll have to spot a duality between dualities one day.
We know they are there, it's maybe more a matter of telling an interesting story about them. Namely given any one explicit equivalence between the quantizations of two different Lagangians, then all elements in the same connected component of its hom-space will be dual dualities.
I expect that if one would write an MO/PO message that briefly explains what we are after, people would recognize some examples. But myself, I don't feel I have time for writing this at the moment.

We’ll have to spot a duality between dualities one day.

We know they are there, it’s maybe more a matter of telling an interesting story about them. Namely given any one explicit equivalence between the quantizations of two different Lagangians, then all elements in the same connected component of its hom-space will be dual dualities.

I expect that if one would write an MO/PO message that briefly explains what we are after, people would recognize some examples. But myself, I don’t feel I have time for writing this at the moment.

Author: David_CorfieldFormat: MarkdownItexThere's a good talk by Vafa [On Mathematical Aspects of String Theory](https://www.youtube.com/watch?v=yreUdrIbt2Q) in which he discusses dualities as 'highly nontrivial isomorphisms'.

Author: David_CorfieldFormat: MarkdownItexThere's a nice paper on dual descriptions in physics recently out by [[Mina Aganagic]], [String Theory and Math: Why This Marriage May Last](http://arxiv.org/abs/1508.06642), so I started a page for her.
Presumably she's alluding to Dyson's 1972 jest:
>I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.

Author: David_CorfieldFormat: MarkdownItexWe were speaking about "Duality of/between dualities" above, and in the entry [[duality in physics]]. I see Duff speaks of "duality of dualities" on p. 26 of [arXiv:hep-th/9805177](https://arxiv.org/abs/hep-th/9805177),
>It occurred to the present author that string/string duality has another unexpected payoff. If we compactify the six-dimensional spacetime on two circles down to four dimensions, the fundamental string and the solitonic string will each acquire a T-duality. But here is the miracle: the T-duality of the solitonic string is just the S-duality of the fundamental string, and vice-versa!
But I guess that's not in the same sense as here.
> It did not take long to realize that 6-dimensional string/string duality (and hence 4-dimensional electric/magnetic duality) follows from 11-dimensional membrane/fivebrane
duality.
Does something of that duality of dualities exist up at the level of 11-dimensional membrane/fivebrane duality?
This from the abstract of [arXiv:0906.3013](https://arxiv.org/abs/0906.3013) sounds like Duff's sense too:
>we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be understood as the simple result that all of these theories are different realizations of one and the same system in M-theory.
and
>Toric duality is Seiberg duality and toric duality is mirror symmetry: a duality of dualities

It occurred to the present author that string/string duality has another unexpected payoff. If we compactify the six-dimensional spacetime on two circles down to four dimensions, the fundamental string and the solitonic string will each acquire a T-duality. But here is the miracle: the T-duality of the solitonic string is just the S-duality of the fundamental string, and vice-versa!

But I guess that’s not in the same sense as here.

It did not take long to realize that 6-dimensional string/string duality (and hence 4-dimensional electric/magnetic duality) follows from 11-dimensional membrane/fivebrane
duality.

Does something of that duality of dualities exist up at the level of 11-dimensional membrane/fivebrane duality?

we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be understood as the simple result that all of these theories are different realizations of one and the same system in M-theory.

and

Toric duality is Seiberg duality and toric duality is mirror symmetry: a duality of dualities

Author: UrsFormat: MarkdownItexFirst time that I see somebody pick up the perspective from the entry _[[duality in physics]]_:
* Yuji Tachikawa, _What is Quantum Field Theory?_ ([pdf](http://indico.ipmu.jp/indico/event/134/contribution/17/material/slides/0.pdf))
from slide 39 on.

Author: David_CorfieldFormat: MarkdownItexOh, I was hoping from that that they'd really picked up our perspective. So nothing on dualities between dualities, and most attention on it not being an epimorphism to all QFTs in
$$
quantization
\;\colon\;
LagrangianData \longrightarrow LagrangianQFTs
\hookrightarrow
QFTs
\,.
$$

Oh, I was hoping from that that they’d really picked up our perspective. So nothing on dualities between dualities, and most attention on it not being an epimorphism to all QFTs in

Author: UrsFormat: MarkdownItex> Oh, I was hoping
One step at a time. Now that they re-discovered that not every field theory is Lagrangian (understood by [Helmholtz in 1887](https://ncatlab.org/nlab/show/Helmholtz%20operator#Helmholtz87)) they are asking for a definition of QFT without a Lagrangian, understood by [Haag in 1959](https://ncatlab.org/nlab/show/AQFT#Haag59).
At this pace, it will take a bit until they get to simplicial sets and dualities between dualities ;-)

Oh, I was hoping

One step at a time. Now that they re-discovered that not every field theory is Lagrangian (understood by Helmholtz in 1887) they are asking for a definition of QFT without a Lagrangian, understood by Haag in 1959.

At this pace, it will take a bit until they get to simplicial sets and dualities between dualities ;-)

Author: David_CorfieldFormat: MarkdownItexYou had the idea (#54) that dualities between dualities might already be out there. The trouble is that the literature often means by that a pair of dualities, so two dual theories which are dual to another pair of dual theories. That seems to be happening in [Hidden Finite Symmetries in String Theory and Duality of Dualities](https://arxiv.org/abs/hep-th/9508132).
Might "symmetries of symmetries" throw something up? But you use that expression to speak of gauge-of-gauge transformations in [Lie n-algebras of BPS charges](https://arxiv.org/abs/1507.08692). I guess in a similar vein, there is [CP and other Symmetries of Symmetries](https://arxiv.org/abs/1608.05240):
>This work is devoted to the study of outer automorphisms of symmetries ("symmetries of symmetries") in relativistic quantum field theories (QFTs). Prominent examples of physically relevant outer automorphisms are the discrete transformations of charge conjugation (C), space–reflection(P), and time–reversal (T).

This work is devoted to the study of outer automorphisms of symmetries (“symmetries of symmetries”) in relativistic quantum field theories (QFTs). Prominent examples of physically relevant outer automorphisms are the discrete transformations of charge conjugation (C), space–reflection(P), and time–reversal (T).

Author: David_CorfieldFormat: MarkdownItexShouldn't our duality story be extended to include non-Lagrangian theories? e.g., as it stands we aren't representing dualities between Lagrangian and non-Lagrangian theories, as in in AdS-CFT.
Is it that we ought to have a larger domain than ‘Lagrangian data’ on which to construct the homotopy quotient?

Shouldn’t our duality story be extended to include non-Lagrangian theories? e.g., as it stands we aren’t representing dualities between Lagrangian and non-Lagrangian theories, as in in AdS-CFT.

Is it that we ought to have a larger domain than ‘Lagrangian data’ on which to construct the homotopy quotient?